Unitary spherical spectrum for p-adic classical groups

Let G be a split reductive p-adic group. Then the determination of the unitary representations with nontrivial Iwahori fixed vectors can be reduced to the determination of the unitary dual of the corresponding Iwahori-Hecke algebra. In this paper we study the unitary dual of the Iwahori-Hecke algebras corresponding to the classical groups. We determine all the unitary spherical representations.     

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.     

Unitarizability of a certain class of irreducible representations of classical groups

We prove that a certain class of irreducible representations of the classical p-adic groups is unitarizable and in general, can be isolated in the unitary dual. These representations are Aubert duals of a certain class of square-integrable representations, thus, in this case, Bernstein’s conjecture, which states that the Aubert involution preserves unitarizability, is confirmed.     

Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C^∗-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.     

Langlands classification for real Lie groups with reductive Lie algebra

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations U^V, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G ₀ is finite, so the center of the connected semi-simple subgroup with Lie algebra [g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).     

On minimal representations of Chevalley groups of type D n , E n and G 2

Let G be a simply connected Chevalley group of type D _n , E _n or G₂. In this paper, we show that the minimal representation of G is unique for types D _n and E _n and it does not exist for the type G₂.     

The Contraction of S2p−1 to Hp−1

 A contraction of the sphere , considered as the homogeneous space , to the Heisenberg group is defined. The infinite dimensional irreducible unitary representations of Heisenberg group are then shown to be the limits of the irreducible representations of which are class-1 with respect to . Our results generalise the earlier results of Fulvio Ricci. (Received 1 July 1998; in revised form 3 November 1998)     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Construction of homogeneous minimal 2-spheres in complex Grassmannians

In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU (2). Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G(2, n + 1) by making use of Veronese sequence.     

Lowest weight representations of some infinite dimensional groups on fock spaces

The results of Kashiwara and Vergne on the decomposition of the tensor products of the Segal-Shale-Weil representation are extended to the infinite dimensional case and give all unitary lowest weight representations. Our methods are basically algebraic. When restricted to the finite dimensional case, they yield a new proof.     

Unitary representations of the groups of measurable and continuous functions with values in the circle

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Stability of the local gamma factor in the unitary case

In [Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291-313, MR2195117 (2006m:22026)], Rallis and Soudry prove the stability under twists by highly ramified characters of the local gamma factor arising from the doubling method, in the case of a symplectic group or orthogonal group G over a local non-archimedean field F of characteristic zero, and a representation π of G, which is not necessarily generic. This paper extends their arguments to show the stability in the case when G is a unitary group over a quadratic extension E of F, thereby completing the proof of the stability for classical groups. This stability property is important in Cogdell, Kim, Piatetski-Shapiro, and Shahidi's use of the converse theorem to prove the existence of a weak lift from automorphic, cuspidal, generic representations of G(A) to automorphic representations of GLn(A) for appropriate n, to which references are given in [Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291-313, MR2195117 (2006m:22026)].     

Compositions of theta correspondences

Theta correspondence θ over is established by Howe (J. Amer. Math. Soc. 2 (1989) 535). In He (J. Funct. Anal. 199 (2003) 92), we prove that θ preserves unitarity under certain restrictions, generalizing the result of Li (Invent. Math. 97 (1989) 237). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method (J. Algebra 190 (1997) 518; Representation Theory of Lie Groups, Park City, 1998, pp. 179-238; The Orbit Correspondence for real and complex reductive dual pairs, preprint, 2001), one can compose theta correspondences to obtain unitary representations. We call this process quantum induction.     

Positive-definite functions on infinite-dimensional groups

This paper concerns positive-definite functions on infinite-dimensional groups G. Our main results are as follows: first, we claim that if G has a σ-finite measure μ on the Borel field whose right admissible shifts form a dense subgroup G ₀, a unique (up to equivalence) unitary representation (H, T) with a cyclic vector corresponds to through a method similar to that used for the G–N–S construction. Second, we show that the result remains true, even if we go to the inductive limits of such groups, and we derive two kinds of theorems, those taking either G or G ₀ as a central object. Finally, we proceed to an important example of infinite-dimensional groups, the group of diffeomorphisms on smooth manifolds M, and see that the correspondence between positive-definite functions and unitary representations holds for under a fairy mild condition. For a technical reason, we impose condition (c) in Sect. 2 on the measure space throughout this paper. It is also a weak condition, and it is satified, if G is separable, or if μ is Radon. This research was partially supported by a Grant-in-Aid for Scientific Research (No.18540184), Japan Socieity of the Promotion of Science.     

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyl's branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples. Partially supported by a grant from the Department of Energy. Partially supported by NSF grant No. MCS81-02345.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

Hardy-Sobolev type inequalities on the H-type group

Motivated by the idea of Badiale and Tarantello who have found Hardy-Sobolev inequalities on Rⁿ, a class of Hardy-Sobolev type inequalities on H-type groups is proved via a new representation formula for functions. Extremal functions realizing equality in the inequalities are discussed by refined Concentration-Compactness principles. Finally, some sharp constants for Hardy type inequalities are given. The project supported by National Natural Science Foundation of China, Grant No. 10371099.